matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv_preserve_sub.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/rt_computation/cpms_fpbg.ma".
  16 include "basic_2/dynamic/cnv.ma".
  17
  18 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
  19
  20 (* Inductive premises for the preservation results **************************)
  21
  22 definition IH_cnv_cpm_trans_lpr (a) (h): relation3 genv lenv term ≝
  23                                 λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  24                                 ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡[n,h] T2 →
  25                                 ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  26
  27 definition IH_cnv_cpms_trans_lpr (a) (h): relation3 genv lenv term ≝
  28                                  λG,L1,T1. ⦃G, L1⦄ ⊢ T1 ![a,h] →
  29                                  ∀n,T2. ⦃G, L1⦄ ⊢ T1 ➡*[n,h] T2 →
  30                                  ∀L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L2⦄ ⊢ T2 ![a,h].
  31
  32 definition IH_cnv_cpm_conf_lpr (a) (h): relation3 genv lenv term ≝
  33                                λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  34                                ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  35                                ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  36                                ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  37
  38 definition IH_cnv_cpms_strip_lpr (a) (h): relation3 genv lenv term ≝
  39                                  λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  40                                  ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 →
  41                                  ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  42                                  ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  43
  44 definition IH_cnv_cpms_conf_lpr (a) (h): relation3 genv lenv term ≝
  45                                 λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
  46                                 ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡*[n1,h] T1 → ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡*[n2,h] T2 →
  47                                 ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
  48                                 ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[n2-n1,h] T & ⦃G, L2⦄ ⊢ T2 ➡*[n1-n2,h] T.
  49
  50 (* Auxiliary properties for preservation ************************************)
  51
  52 fact cnv_cpms_trans_lpr_sub (a) (h):
  53                             ∀G0,L0,T0.
  54                             (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_trans_lpr a h G1 L1 T1) →
  55                             ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_trans_lpr a h G1 L1 T1.
  56 #a #h #G0 #L0 #T0 #IH #G1 #L1 #T1 #H01 #HT1 #n #T2 #H
  57 @(cpms_ind_dx … H) -n -T2
  58 /3 width=7 by fpbg_cpms_trans/
  59 qed-.
  60
  61 fact cnv_cpm_conf_lpr_sub (a) (h):
  62                           ∀G0,L0,T0.
  63                           (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
  64                           ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpm_conf_lpr a h G1 L1 T1.
  65 /3 width=8 by cpm_cpms/ qed-.
  66
  67 fact cnv_cpms_strip_lpr_sub (a) (h):
  68                             ∀G0,L0,T0.
  69                             (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_conf_lpr a h G1 L1 T1) →
  70                             ∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h] ⦃G1, L1, T1⦄ → IH_cnv_cpms_strip_lpr a h G1 L1 T1.
  71 /3 width=8 by cpm_cpms/ qed-.